Question

Let X be the number of distinct 16-bit integers in 2’s complement representation. Let Y be the number of distinct 16-bit integers in sign magnitude representation.

Then X – Y is _______.

Answer 1

For n bits,

Total number of integers in 2’s complement representation = -2^n-1 - 1 to 2^n-1

Total number of integers in sign magnitude representation = - (2^n-1 - 1) to (2^n-1 -1)

In case of 16- bit,

Total distinct integers in 2’s complement representation (X)= -2^16-1 - 1 to 2^16-1 = -2¹⁵ - 1 to 2¹⁵

Total number of integers in sign magnitude representation (Y)= - (2^16-1 - 1) to (2^16-1 - 1) = -2¹⁵ - 1 to 2¹⁵ – 1

So, difference between X and Y = 1

Because in sign magnitude representation, there are two representation for 0 as +0 and – 0.